Topology - Topology Topics - Some Theorems in General Topology

Some Theorems in General Topology

• Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
• Every continuous image of a compact space is compact.
• Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
• A compact subset of a Hausdorff space is closed.
• Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
• Every sequence of points in a compact metric space has a convergent subsequence.
• Every interval in R is connected.
• Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.
• The continuous image of a connected space is connected.
• Every metric space is paracompact and Hausdorff, and thus normal.
• The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
• The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
• Any open subspace of a Baire space is itself a Baire space.
• The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
• On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
• Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.

General topology also has some surprising connections to other areas of mathematics. For example:

• In number theory, Fürstenberg's proof of the infinitude of primes.

See also some counter-intuitive theorems, e.g. the Banach–Tarski paradox.